Numerical Solutions of Laplace's Equation for Various Physical Situations
There are two projects in this thesis. In the first project, a general method is introduced to numerically calculate the resistance of truncated resistors in cylindrical coordinates, with non-constant cross-sectional area. The problem of finding the resistance of a truncated conical resistor is given in some introductory textbooks as a simple problem. The textbook method is flawed however, and leads to the wrong answer. The textbook method assumes that the electric potential distribution inside the truncated cone is approximately equivalent to a cylindrical resistor. This assumption ignores the constricting affect that the boundary of the truncated conical resistor has on the electric potential inside. The deformation of the electric field is not accounted for by excess charge or changing magnetic fields, instead it is the result of a derivative operation called the shear of the field. Numerical solutions for the resistance of truncated conical, ellipsoidal, and hyperboloidal resistors are presented as a function of a/b, where a is the radius of the smallest cross-sectional area and b is radius of the largest. It was found that the textbook solution always underestimates the numerical value of the resistance. In the second project, dielectric breakdown clusters were grown with a stochastic two dimensional Dielectric Breakdown Model (DBM) on a honeycomb, square, and triangle lattice, as well as on a random distribution of nodes. On the regular lattices the number of nearest neighbours was a constant at all lattice sites. For a random distribution of nodes there was variation in the number of nearest neighbours at different nodes. Some percentage of the nodes were isolated from the rest of the distribution, because they had 0 nearest neighbours. Distributions of nodes in which many of the nodes had 0 nearest neighbours indicated a medium with high density fluctuations. The motivation for this work was to study the relationship between the fractal dimension of the dielectric breakdown clusters and the number of nearest neighbours, and the density variation of the medium. The singularity spectra were calculated for the clusters, as well as their fractal dimension using box counting, and sandbox methods. It was found that the dielectric breakdown model produces monofractal clusters. As such, the dimension of the clusters can be represented by a single fractal dimension. In the DBM, the probability of a perimeter site connecting to the cluster is proportional to the strength of the local electric field raised to an exponent. If the exponent is a large positive number then perimeter sites which feel a stronger electric field are more likely to connect to the cluster. Increasing the exponent produces clusters which resemble lightning, with a fractal dimension lower than the dimension of the lattice. Similarly increasing the percentage of isolated nodes decreases the fractal dimension.